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Spaces of generalized splines over T-meshes

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DOI:http://dx.doi.org/10.1016/j.cam.2015.08.006 Terms of use:

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This is the author’s final version of the contribution published as:

Cesare Bracco and Fabio Roman.

Spaces of generalized splines over

T-meshes. Journal of Computational and Applied Mathematics, Volume 294,

2016, pp. 102-123, DOI:10.1016/j.cam.2015.08.006

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Spaces of generalized splines over T-meshes

Cesare Braccoa,∗, Fabio Romana

aDepartment of Mathematics “G. Peano”- University of Turin

V. Carlo Alberto 10, Turin 10123, Italy Tel.: +39-011-6702827 Fax: +39-011-6702878

Abstract

We consider a class of non-polynomial spaces, namely a noteworthy case of Extended Chebyshev spaces, and we generalize the concept of polynomial spline space over T-mesh to this non-polynomial setting: in other words, we focus on a class of spaces spanned, in each cell of the T-mesh, both by polynomial and by suitably-chosen non-polynomial functions, which we will refer to as generalized splines over T-meshes. For such spaces, we provide, under certain conditions on the regularity of the space, a study of the dimension and of the basis, based on the notion of minimal determining set, as well as some results about the dimension of refined and merged T-meshes. Finally, we study the approximation power of the just constructed spline spaces.

Keywords: T-mesh, Generalized splines, Dimension formula, Basis functions, Approximation power 2010 MSC: 41A15 (Spline approximation), 65D07 (Splines)

1. Introduction

The theory of Chebyshevian and Quasi-Chebyshevian spline spaces is a well-known tool which allows to generalize the classical concept of univariate polynomial spline spaces to a non-polynomial setting (see, e.g., [1] and [2]). Essentially, the elements of such spaces locally belong to Extended Chebyshev and Quasi-Extended Chebyshev spaces (see, e.g., [2]), respectively. Many papers considered particular cases of Chebyshevian and Quasi-Chebyshevian splines (see, e.g., [3], [4] and [5]).

This paper deals with the application of the concept of spline space over T-meshes to the noteworthy case of the Extended Chebyshev spaces considered in [6], in order to get a generalization of the polynomial spline spaces over T-meshes. The idea of spline spaces over T-meshes was first introduced for polynomial splines by Deng et al. in [7] and further studied by the same authors and several others (see, e.g., [8], [9], [10] and [1]). The basic idea consists of considering spline functions which are polynomials of a certain degree in each of the cells of the T-mesh, which, unlike the classical tensor-product meshes, allows T-junctions, that is, vertices where only three edges meet. This structure, unlike the one of tensor-product meshes, allows the use of local refinement techniques, and for this reason has gathered a lot of attention in the scientific community, which brought to the study not only of spline spaces over T-meshes, but also of the closely-related T-splines (see, e.g., [11], [12], [13] and [14]), the hierarchical splines (see, e.g., [15] and [16]), and the LR-splines (see, e.g., [17]). Our goal is then using the generalized splines of type [6] to define a class of spaces of non-polynomial splines over T-meshes. The relevance of this class of spline spaces and some of the basic concepts related to it have been recently discussed in some international conferences. The study of these non-polynomial spaces is justified by at least two reasons. First of all, the presence of non-polynomial functions allows to exactly reproduce certain shapes which can only be approximated by polynomial splines or NURBS (for example relevant curves like helices, cycloids, catenaries, or other transcendental curves). Moreover, as we will also point out in Section 4, choosing suitable non-polynomial functions also allows an easier computation of derivatives and integrals of certain surfaces with respect to using NURBS (see also [18], [4]). For these reasons, the same kind of non-polynomial functions have been recently used also to construct non-polynomial T-splines (see, e.g., [19]), and non-polynomial hierarchical splines spaces (see [20]). The goal of this work is to carry out a rigorous and deep study of this class of splines, which we will

Corresponding author; moved to Department of Mathematics “U. Dini”- University of Florence: V.le Morgagni 67/a, Florence 50134, Italy Email addresses: cesare.bracco@unito.it (Cesare Bracco), fabio.roman@unito.it (Fabio Roman)

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call Generalized splines over T-meshes, including results about the space dimension and the approximation power, which, as far as we know, are still missing in the literature.

Starting from some of the results obtained in [6] about a noteworthy class of univariate non-polynomial spaces, we define generalized spline spaces over T-meshes and construct a local representation in the Bernstein-B´ezier fashion for their elements. For the above spaces we first provide the construction of a basis and a dimension formula by using the properties of the local Bernstein-B´ezier representation and by generalizing to the non-polynomial case some of the techniques proposed for the polynomial one in [1]. We also analyze how the dimension of such spaces changes when we refine the T-mesh and when we merge two T-meshes.

Moreover, we also study the approximation power of the just constructed spline spaces. In particular, we do it by constructing a quasi-interpolant based on some new local approximants, whose construction is not trivial. In fact, the results about the univariate non-polynomial Hermite interpolants given in [6] cannot be directly extended to the bivariate case. On the other hand, also the bivariate averaged Taylor expansions used in [1] cannot be simply adapted to the non-polynomial case we consider here. Therefore, we instead defined a new local Hermite interpolant belonging to the non-polynomial spline space, whose existence is proved by using certain assumptions made about the non-polynomial functions spanning the space, as carefully explained in Section 4. This approach allows us to get, at least in certain cases, the same approximation order as in the polynomial case.

The paper is organized as follows. Section 2 includes several preliminary arguments about the non-polynomial spaces we will use to define the new spline spaces, including some important properties about the derivatives of the basis functions and the basic concepts about T-meshes. Section 3 presents the new generalized spline spaces over T-meshes, and includes a detailed proof of the dimension formula and of the construction of the basis; moreover, we also provide a study of how the spline space dimension changes when the T-mesh is refined, and of the dimension of a generalized spline space over two merged T-meshes. Section 3 also includes some examples of basis functions, with some remarks about their features. Finally, Section 4 is devoted to the study of the approximation power of the constructed generalized spline space.

2. Preliminaries

The spaces we will consider are of the type

Pn

u,v([a, b]) := h1, s, ..., sn−2, u(s), v(s)i, s∈ [a, b], 2≤ n ∈ IN, (1)

where u, v ∈ Cn+1([a, b]); for n = 1 we set

P1

u,v([a, b]) := hu(s), v(s)i, s∈ [a, b].

We assume that dim Pun,v([a, b]) = n + 1; moreover, in order to prove some of the properties we are about to present, we will sometimes require the following additional conditions on Pn

u,v([a, b]) ∀ψ∈ Pn u,v([a, b]), ifψ(n−1)(s1) =ψ(n−1)(s2) = 0, s1, s2∈ [a, b], s16= s2 thenψ(n−1)(s) = 0, s∈ [a, b]; (2) ∀ψ∈ Pun,v([a, b]), ifψ(n−1)(s1) =ψ(n)(s1) = 0, s1∈ (a, b), thenψ(n−1)(s) = 0, s∈ [a, b]. (3)

In the following, we will explicitly mention when such conditions are needed. 2.1. Normalized positive basis and its properties

In this subsection we consider a normalized positive basis for the space Pun,v([a, b]). The procedure to obtain

it and its fundamental properties are known and can be found in [6]. Therefore here we will just recall the main results obtained in [6], omitting the proofs. We will instead prove Property 2, which will be crucial in order to obtain some results later in the paper.

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The normalized positive basis can be constructed by using the following integral recurrence relation. By (2), there exist unique elements U0,1,nand U1,1,nbelonging tohu(n−1), v(n−1)i satisfying

U0,1,n(a) = 1, U0,1,n(b) = 0,

U1,1,n(a) = 0, U1,1,n(b) = 1, (4)

and

U0,1,n(s),U1,1,n(s) > 0, s∈ (a, b). (5)

Moreover, we define, for k= 2, ..., n and n ≥ 2

U0,k,n(s) := 1 −V0,k−1,n(s) Ui,k,n(s) := Vi−1,k−1,n(s) −Vi,k−1,n(s), 1≤ i ≤ k − 1 Uk,k,n(s) := Vk−1,k−1,n(s), (6) where Vi,k,n(s) := Z s a Ui,k,n/di,k,ndt, (7) and di,k,n(s) := Z b a Ui,k,ndt,

for i= 0, ..., k, k = 1, ..., n − 1. Note that (4) and (5) hold also in the particular case n = 1, and then U0,1,1and U1,1,1

are a positive basis for Pu1,v([a, b]). The following results can be proved about the just defined functions.

Theorem 1. For k= 2, ..., n and n ≥ 2, the set of functions {U0,k,n, ...,Uk,k,n} is a basis for the space h1, s, ..., sk−2, u(n−k)(s), v(n−k)(s)i.

Moreover, it is a normalized positive basis, that is, satisfies the conditionski=0Ui,k,n(s) = 1 and Ui,k,n(s) > 0 for

s∈ (a, b), i = 0, ..., k.

Corollary 1. The set of functions{U0,n,n, ...,Un,n,n} is a normalized positive basis for the space Pun,v([a, b]), n ≥ 2,

Ui,n,n= Bi,n, where{Bi,n}in=0 satisfyni=0Bi,n(s) = 1 and Bi,n(s) > 0 for s ∈ (a, b), i = 0, ..., n. For n = 1, the set

{U0,1,1,U1,1,1} is a positive basis of Pu1,v([a, b]).

Since in the case n= 1 we cannot, in general, guarantee the construction of a normalized positive basis, in the

following we will assume n≥ 2. As a consequence of the results given in Sections 4 and 6 of [6], we get the

following property.

Property 1. For i= 0, ..., k, k = 2, ..., n and n ≥ 2, we have

Ui( j),k,n(a) = 0, j= 0, ..., i − 1,

Ui( j),k,n(b) = 0, j= 0, ..., k − i − 1.

In particular, if we consider k= n, we have

B( j)i,n(a) = 0, j= 0, ..., i − 1,

B( j)i,n(b) = 0, j= 0, ..., n − i − 1.

Property 2. For k= 2, ..., n and n ≥ 2, we have

Ui(i),k,n(a) 6= 0, i= 0, ..., k − 1, (8)

Ui(k−i),k,n (b) 6= 0, i= 1, ..., k. (9)

In particular, if we consider k= n, we have

B(i)i,n(a) 6= 0, i= 0, ..., n − 1,

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Proof. First, let us prove (8) by induction. For k= 2, (8) holds, since from (4), (6) and (7) we get U0,2,n(a) = 1 −V0,1,n(a) = 1 − Z a a U0,1,n(t)/d0,1,ndt= 1 − 0 = 1, U1(1),2,n(a) = D[V0,1,n(s) −V1,1,n(s)]s=a= U0,1,n(a) d0,1,nU1,1,n(a) d1,1,n = 1 d0,1,n − 0 6= 0.

Now, if (8) holds for k, it must be true for k+ 1 as well, since we have

U0,k+1,n(a) = 1 −V0,k,n(a) = 1 − Z a a U0,k,n(t)/d0,k,ndt= 1 − 0 = 1, Ui(i),k+1,n(a) =U (i−1) i−1,k,n(a) di−1,k,nU (i−1) i,k,n (a) di,k,n =U (i−1) i−1,k,n(a) di−1,k,n 6= 0,

where we used (6), (7), Property 1 and the induction hypothesis. Analogously we can prove (9). 

Note that the above constructed basis is not only normalized positive, but it is also a Bernstein basis. 2.2. Some definitions on T-meshes

We will now recall the definition of T-mesh and of some related objects, using the notations of [1]. Note that the concept of T-mesh we will consider here may slightly differ from other ones in the literature, such as the more general used in [21], which allows the presence not only ofT-junctions, but ofL-junctionsandI-junctionsas well.

Definition 1. A T-mesh is a collection of axis-aligned rectangles= {Ri}Ni=1 such that the domainΩ≡ ∪iRi is connected and any pair of rectangles (which we will callcells) Ri, Rj∈∆intersect each other only at points on their edges.

Note that this definition does not imply that the domain Ωis rectangular and allows the presence of holes in it. Tensor-product meshes are a particular case of T-meshes. If a vertex v of a cell belonging to∆lies in the interior of an edge of another cell, then we call it aT-junction.

Definition 2. Given a T-mesh∆, a line segment e connecting the vertices w1and w2is callededge segmentif there

are no vertices lying in its interior. Instead, if all the vertices lying in its interior are T-junctions and if it cannot be extended to a longer segment with the same property, then we call it acomposite edge.

In the following, we will consider T-meshes which areregularand have nocycles, in the sense of the following definitions (see [1] for more details).

Figure 1:An example of regular T-mesh. Figure 2:An example of non-regular T-mesh.

Definition 3. A T-meshis regular if for each of its vertices w the set of all rectangles containing w has a connected interior.

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See Figures 1-2 for examples of regular and not regular T-meshes.

Definition 4. Let w1, ..., wn be a collection of T-junctions in a T-meshsuch that wi lies in the interior of a composite edge having one of its endpoints at wi+1 (we assume wn+1= w1). Then w1, ..., wnare said to form a cycle.

See Figure 3 for an example of cycle in a T-mesh.

Figure 3:The sequencew1, w2, w3, w4is a cycle.

3. Spaces of generalized splines on T-meshes

In this Section, we define the spaces of generalized splines over T-meshes, and we study their dimension by constructing a basis. The results obtained can be considered a generalization to non-polynomial splines spaces over T-meshes of the ones proved in [1] for the basic polynomial case.

3.1. Basics

Let∆be a regular T-mesh without cycles, and let 0≤ r1< n1, 0≤ r2< n2, where r1, r2, n1, n2are integers and

n1, n2≥ 1. We will use the notation r = (r1, r2) and n = (n1, n2).

We define the space of generalized splines over the T-mesh∆of bi-degree n and smoothness r, GSnu,r,v(∆), as

GSnu,r,v(∆) := {p(s,t) ∈ Cr() : p|

R∈ PunR,vR(R) ∀R ∈∆}, (10)

whereΩ= ∪R∈∆R, Cr(Ω) denotes the space of functions p such that their derivatives DisD j

tp are continuous for all 0≤ i ≤ r1and 0≤ j ≤ r2, and the space PunR,vR(R) is defined as

Pn uR,vR(R) := P n1 uR 1,vR1 ([aR, bR]) ⊗ PunR2 2,vR2 ([cR, dR]), (11)

with R := [aR, bR] × [cR, dR], and uR= (uR1, uR2) and vR= (vR1, vR2) such that uR1, vR1 ∈ Cn1+1([aR, bR]), u2, v2∈

Cn2+1([c

R, dR]), dim PunR1 1,vR1

([aR, bR]) = n1+ 1, dim PunR2 2,vR2

([cR, dR]) = n2+ 1, and satisfying both (2) and (3).

In other words, GSnu,r,v(∆) is a space of spline functions which, restricted to each cell R, are products of functions

belonging to spaces of type (1).

We introduce now on each cell R a Bernstein-B´ezier representation for the elements of GSnu,r,v(∆) based on the

Bernstein basis of Pn1 uR1,vR

1

([aR, bR]) and PunR2 2,vR2

([cR, dR]) constructed in Theorem 1; therefore, we need to assume that (2) is satisfied both by Pn1

uR 1,vR1 ([aR, bR]) and PunR2 2,vR2 ([cR, dR]). Let us denote by {BRi,n1} n1 i=0and{BRj,n2} n2 j=0the

Bernstein basis of, respectively, Pn1 uR

1,vR1

([aR, bR]) and PunR2 2,vR2

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coordinates aR, bR, cR, dRof the vertices of the cell R. For any p∈ GSnu,r,v(), we can then give on the cell R the following representation: p|R(s,t) = n1

i=0 n2

j=0 cRi jBRi,n 1(s)B R j,n2(t), (12)

where cRi j∈ IR are suitable coefficients. Let us define the set ofdomain points associated to R:

Dn,R:= {ξi jR}n1,n2 i=0, j=0, with ξR i j:= (n1− i)aR+ ibR n1 ,(n2− j)cR+ jdR n2  , i= 0, ..., n1, j = 0, ..., n2.

We can then define theset of domain pointsfor a given T-mesh∆as

Dn,:= [

R∈∆ Dn,R,

where we assume that multiple appearances of the same point are allowed. If we set BRξ(s,t) := BRi,n1(s)B

R

j,n2(t), whereξ R i j:=ξ, then, for each R∈∆, we can re-write (12) in the more compact form

p|R(s,t) =

ξ∈Dn,R

cξBRξ(s,t),

which we callBernstein-B´ezier form; we refer to the cRξ as theB-coefficients. It is then clear that any element of the space GSnu,r,v() is completely determined by a set of B-coefficients {cξ}ξ∈Dn,∆. Of course, not every choice of the B-coefficients corresponds to an element in the spline space, since smoothness conditions must be satisfied. 3.2. Smoothness conditions

In order to study the consequences of the smoothness conditions required for GSnu,r,v(∆) on the determination of

the B-coefficients of an element of the space, first we need to recall some more concepts about domain points. Let w be the bottom-left vertex of a cell R, andµ:= (µ1,µ2) withµ1≤ n1andµ2≤ n2. We call the set DµR(w) :=

i j}iµ=0, j=01,µ2 thedisk of sizeµ aroundw. The disks around the other vertices of R can be defined analogously.

Moreover, we say that the pointsξi jRwith 0≤ i ≤ν lie within a distanceνfrom the edge e= {aR} × [cR, dR] and we use the notation di jR, e) ≤ν. Analogous notations hold for the other edges of R.

Moreover, we can define the set of domain points

Dµ(w) := [

R∈∆w

DµR(w),

where ∆w⊂∆contains only the cells having w as one of their vertices and multiple appearances of a point are allowed in the union. Given a composite edge e, an edge ˜e lying on e and a domain pointξof a cell which has ˜e as one of its edges, if d, ˜e) ≤ν, then we write that d, e) ≤νas well.

The following lemma is a key step to be able to understand the influence of the smoothness conditions around a vertex, and it is analogous to Lemma 3.3 in [1].

Lemma 1. Let p∈ GSun,r,v() and let w be a vertex of∆. Let us consider two cells R and ˜R with vertices (in

counter-clockwise order) w, w2, w3, w4and w, w5, w6, w7, respectively. If the coefficients cξ,ξ∈ DrR(w) are given, then the

coefficients cη,η∈ DR˜

r(w) are uniquely determined by the smoothness conditions at w.

Proof. Let us assume that R and ˜R are like in Figure 4 (the proof for other configurations is analogous). Then, since we have regularity r= (r1, r2) at w, by using Property 1 we get

h

i=0 k

j=0 cRi j˜DhsBRi˜,n 1(aR˜)D k tB ˜ R j,n2(cR˜) = n1

i=n1−h n2

j=n2−k cRi jDhsBRi,n 1(bR)D k tBRj,n2(dR), (13)

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R ˜ R w w2 w3 w4 w5 w6 w7

Figure 4:The common vertexwshared byRandR, with the notation of Lemma 1.˜

for h= 0, ..., r1, k= 0, ..., r2. By using Property 2 it can be shown that the system composed of equations (13),

with a suitable re-ordering of the equations, is lower triangular, which proves the lemma. 

After having studied the influence of smoothness around a vertex, we now study the situation around edges. The two following lemmas can be considered generalizations of Lemma 3.5 and Lemma 3.6 in [1]. However, note that in our nonpolynomial setting Lemma 3.6 cannot be directly generalized, since having different(uR

1, uR2), (vR1, vR2)

in neighbouring cells leads to configurations which are significantly different from the polynomial case, as we will explain in the proof of Lemma 3. Given an edge e, we will use the following notation:

re:= ( r1, if e is vertical, r2, if e is horizontal, De:= ( Ds, if e is vertical, Dt, if e is horizontal, ne:= ( n2, if e is vertical, n1, if e is horizontal,

{(ae, ce), (be, ce)} := coordinates of the endpoints of e,e= {R ∈: R∩ int(e) 6= /0} uRe := ( uR2, if e is vertical, uR1, if e is horizontal, vRe := ( vR2, if e is vertical, vR1, if e is horizontal.

Moreover, we will assume that for any R∈∆and any edge e such that R∈∆e, uRe, vRe are such that dim Pne

uR

e,vRe([ae, be]) = ne+ 1. (14)

Lemma 2. Let e be a composite edge of. Given p∈ GSun,r,v(∆), for any 0 ≤ j ≤ re, Dejp|eis a univariate function belonging to \

R∈∆e

Pne

uR

e,vRe([ae, be]).

Proof. Let us consider a horizontal composite edge e with endpoints w1= (ae, ce) and w5= (be, ce) like the one showed in Figure 5, composed of the edges e1, e2, e3. First, p|R1(s, dR1) gives the values of p both on e1and e2, because both the edges belong to the same cell R1; similarly, p|R2(s, cR2) gives the values of p both on e2and e3, since they belong to R2.

Since p|e2 belongs to P n1

uR11 ,vR11 ([aR1, bR1]) ∩ P

n1

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R1

R2

e1 e2 e3

◦ ◦

w1 w5

Figure 5: The cells considered in the proof of Lemma 2.

p|e2 and p|e3coincide. These arguments can be extended to an arbitrary number of segments in a composite edge, to the case of vertical composite edges, and to derivatives of any order up to re.  Let us now consider a composite edge with endpoints w1and w5, a cell Rewith vertices w1, w2, w3, w4, and another

cell ˜Re with vertices w5, w6, w7, w8. Moreover we assume that w4and w6 lie on e as well (the other cases are

analogous). Let us define

Mek:=      n ξRe i j on1,n2−r2−k i=n1−r1, j=r2+1 , if e is vertical, n ξRe i j on1−r1−k,n2 i=r1+1, j=n2−r2 , if e is horizontal. , k= 1, 2, 3. (15)

Moreover, we will use ˜reto denote r− (1, 0) if e is horizontal, r − (0, 1) if e is vertical, and ˆreto denote r− (2, 0)

if e is horizontal, r− (0, 2) if e is vertical. We also define, for every e:

de:= dim

\

R∈∆e

huRe, vRei.

Lemma 3. Let e be a composite edge of the T-meshwith endpoints we,aand we,b. Let us assume that there

exists a basis satisfying Properties 1 and 2 for the space \ R∈∆e

Pne

uR

e,vRe([ae, be]). Then, the B-coefficients of a spline

p∈ GSnu,r,v() associated to domain pointsξ such that d, e) ≤ reare uniquely determined by the coefficients of p corresponding to the domain points belonging to one the following sets:

• if de= 2,e1,0; • if de= 1,e1,1ore2,0; • if de= 0,e1,2, ore2,1, ore3,0; where ˜ Me1,0 := DRe r (we,a) ∪ D ˜ Re r (we,b) ∪ Me1, ˜ M1,1 e := DrRe(we,a) ∪ D ˜ Re ˜re(we,b) ∪ M 1 e, ˜ M2,0 e := DrRe(we,a) ∪ DrR˜e(we,b) ∪ Me2, ˜ M1,2 e := DrRe(we,a) ∪ D ˜ Re ˆre(we,b) ∪ M 1 e, ˜ M2,1 e := DrRe(we,a) ∪ D ˜ Re ˜re(we,b) ∪ M 2 e, ˜ M3,0 e := DrRe(we,a) ∪ DrR˜e(we,b) ∪ Me3.

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R Re ˜ Re w1 w4 w3 w2 w5 w8 w7 w6 ˆ w1 z1 z4 wˆ5 z3 z2

Figure 6: The cells considered in the proof of Lemma 3.

Proof. Let us consider a horizontal composite edge e as in Figure 6, with endpoints we,a= w1and we,b= w5(the

proof is analogous for the vertical case). Let R∈∆e, and let us denote its vertices of by z1, z2, z3, z4, with z2and z3

lying on e. We will show that the B-coefficients corresponding to the domain pointsξ belonging to Dn,Rand such

that d, e) ≤ r2are uniquely determined.

Let p∈ GSun,r,v(∆), and let us consider integers k and ℓ such that 1 ≤ k ≤ 3, 0 ≤ ℓ ≤ 2, k +ℓ ≤ 3. First of all, assuming

that the B-coefficients corresponding to the domain points inM˜ek,ℓare given, we can compute the derivatives

n DisDtjp(w1) on1−r1−k,r2 i=0, j=0 , n DisDtjp(w5) or1−ℓ,r2 i=0, j=0. (16)

In fact, by Property 1, the computation of these derivatives involves just the B-coefficients contained inM˜ek,ℓ.

Note that, by Lemma 2, we know that Dtjp|e, j= 0, ..., r2, belongs to the univariate space

\ R∈∆e Pn1 uR 1,vR1 ([w1, w5]).

If de= 2, then we set (k, ℓ) = (1, 0), and the proof is analogous to the polynomial case. If de= 1, by differentiating i times with respect to s, and by considering the basis of

\ R∈∆e Pn1 uR1,vR 1 ([w1, w5]) on e

satisfying Properties 1 and 2, denoted by{Bk}nk1=0−1, we can write

Dis n1−1

k=0 ak, jBk(s) = DisD j tp(s,t)|e, j= 0, ..., r2, (17)

If we assume to have the coefficients associated with the elements ofM˜e1,0, we can use the values (16) of derivatives

in w1and w5to determine ak, j’s from the(n1+ 1)(r2+ 1) conditions

           Dis n1−1

k=0 ak, jBk(w1) = DisD j tp(w1)|e i= 0, . . . , n1− r1− 1, Dis n1−1

k=0 ak, jBk(w5) = DisD j tp(w5)|e i= 0, . . . , r1, , j= 0, . . . , r2.

For example, by considering j= 0, we obtain a linear system whose matrix is of the form

A=               D0sB0(w1) D0sB1(w1) . . . D0sBn1−r1−1(w1) D 0 sBn1−r1(w1) . . . D 0 sBn1−1(w1) D1sB0(w1) D1sB1(w1) . . . D1sBn1−r1−1(w1) D 1 sBn1−r1(w1) . . . D 1 sBn1−1(w1) . . . . Dk1 s B0(w1) Dks1B1(w1) . . . Dks1Bn1−r1−1(w1) D k1 s Bn1−r1(w1) . . . D k1 s Bn1−1(w1) Dr1 s B0(w5) Drs1B1(w5) . . . Drs1Bn1−r1−1(w5) D r1 s Bn1−r1(w5) . . . D r1 s Bn1−1(w5) Dk2 s B0(w5) Dks2B1(w5) . . . Dks2Bn1−r1−1(w5) D k2 s Bn1−r1(w5) . . . D k2 s Bn1−1(w5) . . . . D0sB0(w5) D0sB1(w5) . . . D0sBn1−r1−1(w5) D 0 sBn1−r1(w5) . . . D 0 sBn1−1(w5)              

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where k1= n1− r1− 1 and k2= r1− 1. By Properties 1 and 2, we know that the matrix A has the following sparsity

structure (we mark with• non-zero entries, and with ◦ entries which could be either zero or nonzero).               • 0 . . . 0 0 0 . . . 0 ◦ • . . . 0 0 0 . . . 0 . . . . ◦ ◦ . . . • 0 0 . . . 0 0 0 . . . • ◦ ◦ . . . ◦ 0 0 . . . 0 • ◦ . . . ◦ . . . . 0 0 . . . 0 0 0 • ◦ 0 0 . . . 0 0 0 0 •              

Note that there is a linear dependence, necessarily in the rows with the• in the same column, corresponding to

derivative of order n1− r1− 1 w.r.t. w1and derivative of order r1w.r.t. w5. So exactly one of them can be removed

in order to obtain a square matrix, which is nonsingular because it is composed of a lower triangular upper part, and of an upper triangular lower part, with nonzero elements on the diagonal.

The same arguments hold for higher values of j, and so we do not need all the coefficients associated with the elements ofM˜e1,0: it is sufficient to know the coefficients of the elements of eithere2,0ore1,1.

If de= 0, then Dtjp|e, j= 0, ..., r2, belongs to the univariate space

\

R∈∆e

Pn1 uR1,vR

1

([w1, w5]) whose basis we denote by

{Bk}nk=01−2. In this case, assuming again to have all data aboutM˜

1,0

e , the matrices of the systems determining the coefficients aj,k’s have the following sparsity structure:

                • 0 . . . 0 0 0 . . . 0 ◦ • . . . 0 0 0 . . . 0 . . . . ◦ ◦ . . . • 0 0 . . . 0 ◦ ◦ . . . ◦ • 0 . . . 0 0 0 . . . • ◦ ◦ . . . ◦ 0 0 . . . 0 • ◦ . . . ◦ . . . . 0 0 . . . 0 0 0 • ◦ 0 0 . . . 0 0 0 0 •                

The linear dependence is between the rows in which are considered derivatives of order n1− r1− 2 and n1− r1− 1

at w1, and derivatives of order r1− 1 and r1at w5. For every j, in order to get a square nonsingular matrix, we can

delete the first two, or the last two, or the two central of these rows. These deletions corresponds respectively to considering only the setsM˜e3,0, ore1,2, ore2,1.

By writing p|Rin its Bernstein-B´ezier form, we get the linear system: n1

i=0 r2

j=0 cRi jDhsBRi,n 1(aR)D k tBRj,n2(cR) = D h sDtkp|R(aR, cR)

where 0≤ h ≤ n1, 0 ≤ k ≤ r2, the unknowns are the cRi j, and the derivatives DhsDktp|R(aR, cR) are known, since we have just determined p|eand its derivatives. By suitably re-ordering the indices (i, j) and (h, k) we obtain, by Property 1, a lower triangular system where the elements on the diagonal are nonzero due to Property 2. The

Lemma is then proved. 

Remark 1. Note that, in order to prove Lemma 2, we do not require that \

R∈∆e

Pn1 uR

1,vR1

([w1,e, w5,e]) has a

Bernstein-like basis (and then we do not require the conditions (2) and (3) for this space): we just need that the basis e satisfies Properties 1 and 2, which is sufficient to guarantee that the derivatives DisDtjp|e, for i= 0, ..., n1and j= 0, ..., r2,

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3.3. Basis and dimension formula

We will prove the construction of the basis, and a dimension formula for GSnu,r,v(∆), provided that n1≥ 2r1+ 3

and n2≥ 2r2+ 3. We recall the meaning of determining set and minimal determining set.

Definition 5. Let M ⊂ Dn,∆. M is a determining set for GSnu,r,v(∆) if for any spline function p belonging to

GSnu,r,v(∆) such that cξ = 0, ∀ξ ∈ M =⇒ p ≡ 0, where for anyξ ∈ M , cξ is the corresponding B-coefficient of p.

Furthermore, M is minimal if no (strict) subset of it satisfy this property.

Let us denote by JNT the set of vertices which are not T-junctions, and by C the set of composite edges of ∆. Moreover, let Ckbe the subset of the composite edges e∈ C such that de= k, k = 0, 1, 2.

For any w in JNT, let Rwbe a cell with an edge ewhaving an endpoint at w and such that it has maximum length among the edges with an endpoint at ew. Moreover, let

Mw := DRw r (w), for any w∈ JNT MR :=  ξR i j n1−r1−1,n2−r2−1 i=r1+1,r2+1 , for any R∈∆ M := [ w∈JNT Mw [ e∈C2 Me1 [ e∈C1 Me2 [ e∈C0 Me3[ R∈∆ MR (18) where Me1, M2 e, Me3, are defined by (15).

The three following results of this subsection are essentially obtained by using arguments analogous to those used in [1] (they can be considered the generalization of Lemma 4.1, Lemma 4.2 and Theorem 4.3 in [1], respectively). However, we will briefly summarize their respective proofs in order to highlight the role played by some crucial assumptions about the absence of cycles in the T-mesh and about the regularity of the spline space.

Theorem 2. The subset of domain points M ⊂ Dn,∆is a determining set for GSun,r,v(∆).

Proof. In order to prove the lemma we need to show that if p∈ GSnu,r,v(), p|R=∑ξ∈Dn,RcξB R

ξ for any R∈∆

with cξ = 0 ∀ξ ∈ M , then p ≡ 0. By hypothesis, for any w ∈ JNT cξ = 0 for all ξ ∈ Mw= DRrw(w), which

implies, by Lemma 1, that cξ = 0 for allξ ∈ Dr(w). Therefore, for any composite edge e with the endpoints in

JNT, by Lemma 3 we have cξ = 0 for allξ such that d, e) ≤ re, since by hypothesis cξ = 0 ∀ξ ∈Se∈C2M

1 e ∪ S e∈C1M 2 e ∪ S e∈C0M 3

e. We determine the B-coefficients associated with the not yet considered domain points by using an iterative procedure consisting of two steps:

1. for each T-junction w on an already considered composite edge, Lemma 1 implies that cξ = 0 ∀ξ ∈ Dr(w);

2. for each composite edge e whose endpoints have been already considered, Lemma 3 implies that cξ = 0 for

allξ such that d, e) ≤ re.

Since the T-meshes has no cycles, this procedure stops after having considered all the vertices and edges. Then, all the B-coefficients corresponding to domain points within a distance refrom any edge e are determined and are zero. The remaining coefficients are zeros as well, since they correspond to domain pointsξ whose distance from

any edge e is greater than re, that is,ξ∈ ∪R∈∆MR. 

Lemma 4. For everyξ ∈ M , there is one and only oneψξ ∈ GSnu,r,v(∆)

γηψξ =δξ,η, η∈ M , (19)

whereδξ,ηis the Kronecker delta and, for anyη∈ Dn,∆,γη: GSnu,r,v() → IR is the functional defined by

γηp= cη, with cηB-coefficient of p associated toη, p∈ GSnu,r,v(∆). (20)

Proof. For anyξ ∈ M ,ψξ can be constructed as follows: we set cηξ,η, and then we determine the remaining

coefficients by using the same procedure as in the proof of Theorem 2. Note that this way to determine coefficients does not lead to inconsistencies, since we assumed n1≥ 2r1+ 3 and n2≥ 2r2+ 3, which implies that the disks of

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Theorem 3. The subset of domain points M⊂ Dn,∆is a minimal determining set for GSnu,v(,r), the set {ψξ}ξ∈M

is a basis for GSnu,r,v(), and

dim(GSnu,r,v(∆)) = (r1+ 1)(r2+ 1)JNT(∆) + (r2+ 1)(n1− 2r1− 1)Ehor2 (∆)

+(r2+ 1)(n1− 2r1− 2)Ehor1 (∆) + (r2+ 1)(n1− 2r1− 3)Ehor0 (∆)

+(r1+ 1)(n2− 2r2− 1)Ever2 (∆) + (r1+ 1)(n2− 2r2− 2)Ever1 (∆)

+(r1+ 1)(n2− 2r2− 3)Ever0 (∆) + (n1− 2r1− 1)(n2− 2r2− 1)N(∆) (21)

where

JNT(∆) := number of vertices ofwhich are not T-junctions, Ei

hor(∆) := number of horizontal composite edges ofwith de= i, Ei

ver(∆) := number of vertical composite edges ofwith de= i, N(∆) := number of cells.

Proof. The set of functionsξ}ξ∈Mis a basis for GSnu,r,v(∆). In fact, (19) implies that they are linearly independent

and therefore dim(GSnu,r,v()) ≥ |M |. On the other hand, since M is a determining set we have dim(GSun,r,v(∆)) ≤ |M |, and therefore we must conclude that dim(GSnu,v,r(∆)) = |M | and that {ψξ}ξ∈M is a basis. Then, M is a

minimal determining set and the formula for dim(GSnu,r,v(∆)) is obtained from (18). 

Remark 2. From the dimension formula of Theorem 3, it is clear that the dimension of the spline space depends

on the dimensions deof the spaces∩R∈∆ehu

R

e, vRei, e ∈ C. In particular, if for any composite edge e, de≥ 1, then we can relax the conditions on regularity and order, that is, it is sufficient to assume that n1≥ 2r1+ 2, n2≥ 2r2+ 2,

instead of n1≥ 2r1+ 3, n2≥ 2r2+ 3. Similarly, if for any composite edge de= 2 holds, we can further relax the above conditions and replace them with n1≥ 2r1+ 1, n2≥ 2r2+ 1, which are exactly the same conditions required

in the polynomial case.

Lemma 5. The elements of the basisψξ,ξ ∈ M , form a partition of the unity.

Proof. For the spline p=∑ξ∈Mψξ we haveγηp= 1 for anyη ∈ M . Note that, since the local Bernstein-like

basis{BR i,n1(s)B

R

j,n2(t)}i=0,...,n1, j=0,...,n2 satisfy the partition of unity, setting all the B-coefficients c R

ξ to 1,ξ ∈ Dn,R,

R∈∆, gives the constant function 1, which belongs to GSun,r,v(∆). In other words,γη1= 1 for anyη∈ Dn,∆. On

the other hand, we know that the B-coefficient associated to the points of the minimal determining set uniquely determine an element of GSnu,r,v(). Then, we must have p =∑ξ∈Mψξ = 1. 

3.4. Examples

Let n= (5, 5), r = (1, 1), an let us consider the T-mesh∆in Figure 7 and the spline spaces over it

S1:= GSnu,r,v(∆) = {p(s,t) ∈ Cr(Ω) : p|R∈ PunR,vR(R) ∀R ∈∆}, (22) uRi= (cosh(3s), cosh(3t)), vRi= (sinh(3s), sinh(3t)), i= 1, 2, ..., 7,

S2:= GSnu,v,r(∆) = {p(s,t) ∈ Cr(Ω) : p|R∈ PunR,vR(R) ∀R ∈∆}, (23) uR1= (cos(1.9s), cos(1.9t)), vR1 = (sin(1.9s), sin(1.9t)),

uR4= (cosh(3s), cosh(3t)), vR4= (sinh(3s), sinh(3t)), uRi= (s4,t4), vRi= (s5,t5), i= 2, 3, 5, 6, 7,

S3:= GSnu,v,r(∆) = {p(s,t) ∈ Cr(Ω) : p|R∈ PunR,vR(R) ∀R ∈∆}, (24) uRi= (s4,t4), vRi= (s5,t5), i= 1, 2, ..., 7.

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R1 R2 R3 R4 R5 R6 R7 • • • • • • ◦ • ◦ • • • • • 0 π24 π π 2 3π 4 π

Figure 7:T-mesh, where black circles represent vertices belonging toJNT and empty circles are T-junctions.

In other words, S1is a generalized spline space locally spanned by hyperbolic and polynomial functions, S2 is

locally spanned by trigonometric and polynomial, hyperbolic and polynomial or only polynomial functions, while S3is a polynomial spline space over the T-mesh∆.

Note that in the three cases Ehor1 (∆) = E1

ver(∆) = 0, but for S1and S3we have Ehor2 (∆) + Ever2 (∆) = 18, Ehor0 (∆) + Ever0 (∆) = 0, while for S2we have Ehor2 (∆) + Ever2 (∆) = 14, Ehor0 (∆) + E

0

ver(∆) = 4. In fact, for S3 there are 4

composite edges with de= 0, that is

segment with endpoints(0, 3/4π) end (3/4π, 3/4π), segment with endpoints(0,π/2) end (π/2,π/2),

segment with endpoints(π/2,π/2) end (π/2, 3/4π), segment with endpoints(π/2, 0) end (π/2,π/2).

Then, by Theorem 3 we get dim(S1) = dim(S3) = 148 and dim(S2) = 132. In all the cases (included the

polyono-mial one, see [1]), the basis functionsψξ,ξ∈ M , can be determined by setting to 1 the B-coefficient corresponding

to one point of the respective minimal determining set M , to 0 the B-coefficients of the other points of M , and then computing the remaining coefficients by using the scheme described in the proof of Theorem 2.

It is worth stressing that, in spite of the different dimension, by (15) and (18) the minimal determining set for S2is a subset of the ones for S1and S3(which coincide). Therefore, for the three cases there are several basis

functions which are associated to the same domain points and can be compared (see Figures 9 and 8). From the actual computation of their values, it is evident that the elements of the basis are not necessarily non-negative (see, for example, the basis functionψ

ξR1 14

shown in Figure 8). Moreover, we observe that some elements of the global basis coincide with elements of a local basis. For example, in Figure 9ψ

ξR4 22

is both an element of the local basis in the cell R4and an element of the global basis.

Finally, let us show another example: we consider the T-mesh∆in Figure 10 and the corresponding spline space GSnu,r,v(∆), with n = (3, 3), r = (1, 1), uR= (cos(s), cos(t)), vR= (sin(s), sin(t)), for any R ∈∆. This example

allows us to show that the basis is not guaranteed to have a local support. In fact, we can observe that, for example, the basis functionψξR2

00

takes non-zero values in all the cells of the T-mesh (see Figure 10).

3.5. T-mesh refinement and merging

Two key features of meshes are the possibility of local refinement and the ability to easily merge two T-meshes (and the corresponding surfaces). We will then discuss how the space dimension changes when we refine a T-mesh and when we merge two T-meshes, using an approach analogous to [7], where such computations were done for the corresponding polynomial spaces.

3.5.1. Edge insertion

While for a tensor-product mesh inserting a new knot (in either direction) means inserting an entire row or columns of knots in the mesh, in T-meshes we can insert a single edge subdividing only one cell into two smaller

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(a) (b)

(c)

Figure 8: The global basis element associated to the domain pointξR1

14 for (a) the space (22), (b) the space (23)

and (c) the polynomial space(24). Note that for all the considered cases it is not a positive function.

cells.

In the following, we assume that (1) and (2) always hold, both before and after a knot insertion. Moreover, at each refinement we will add a new edge splitting an existing cell into two parts, but we will not introduce any new non-polynomial functions: in the two new cells the considered non-polynomial functions are the same as in the original cell, so that, globally, the new spline space contains the previous one (the spaces are nested).

We consider three possible cases of edge insertion.

• Case (a) (see Figure 11(a)). The edge insertion adds two new T-junctions and one new composite edge

(the inserted edge itself). Since in the dimension formula (21) only the number of vertices which are not T-junctions is used (JNT(∆)), the new vertices do not produce any change in the dimension, while the new composite edge does. Note that for such composite edge de= 2, since we assumed that the refinement generates nested spaces. Then, if we denote by ˜∆the T-mesh obtained by inserting the edge in∆, we have that, if the inserted edge is horizontal,

dim(GSnu,r,v( ˜∆)) = dim(GSnu,r,v(∆))

+(n2− 2r2− 1)(n1− 2r1− 1) + (r2+ 1)(n1− 2r1− 1),

while, if the edge inserted is vertical, dim(GSn,r

u,v( ˜∆)) = dim(GSnu,r,v(∆))

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(a) (b)

(c)

Figure 9: In (a) the elementψξR4 22

of the basis of the space (22)coincides with an element of the local Bernstein-B´ezier basis. The same behaviour holds for the corresponding element of the basis of (b) the space (23)(and therefore the basis function is exactly the same, since the two spaces inR4are spanned by the same functions) and of (c) the polynomial spline space(24).

• Case (b) (see Figure 11(b)). The edge insertion adds one new junction, one new vertex which is not a

T-junction and one new composite edge (the inserted edge itself). Moreover, note that the inserted edge splits into two new edges an edge in the opposite direction: the values of defor these two parts after splitting could be different from the value of de for the original edge (they could not be lower, but they could be higher; see the example of Figure 12(a)). Let∆(Σd) denote the difference between the sum of the values of deafter splitting, and the value of debefore. Then, if the inserted edge is horizontal, we have

dim(GSn,r

u,v( ˜∆)) = dim(GSnu,r,v(∆))

+(n2− 2r2− 1)(n1− 2r1− 1) + (r2+ 1)(n1− 2r1− 1)

+(r1+ 1)(n2− 2r2− 3 +∆(Σd)) + (r1+ 1)(r2+ 1)

while, if the edge inserted is vertical,

dim(GSnu,r,v( ˜∆)) = dim(GSnu,r,v(∆))

+(n1− 2r1− 1)(n2− 2r2− 1) + (r1+ 1)(n2− 2r2− 1)

+(r2+ 1)(n1− 2r1− 3 +∆(Σd)) + (r1+ 1)(r2+ 1)

• Case (c) (see Figure 11(c)). The edge insertion adds two new vertices which are not T-junctions. Moreover,

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0 0.5 1 1.5 2 0 0. 5 1 1. 5 2 w1 w2 w3 w4 w5 w6 w7 w8 w9

R

1

R

2

R

3

R

4

R

5

Figure 10:Sign ofψξR2 00

: at evaluation points marked with a squareψξR2 00

takes positive values, at evaluation points marked with a circle it takes negative values, and at evaluation points marked with a triangle it is zero.

value of de(see Figure 12(b)). If the inserted edge is horizontal we have dim(GSnu,r,v( ˜∆)) = dim(GSnu,r,v(∆))

+(n2− 2r2− 1)(n1− 2r1− 1) + (r2+ 1)(n1− 2r1− 1)

+(r1+ 1)(2n2− 4r2− 6 +∆(Σd)) + 2(r1+ 1)(r2+ 1)

while, if the edge inserted is vertical

dim(GSnu,r,v( ˜∆)) = dim(GSnu,r,v(∆))

+(n1− 2r1− 1)(n2− 2r2− 1) + (r1+ 1)(n2− 2r2− 1)

+(r2+ 1)(2n1− 4r1− 6 +∆(Σd)) + 2(r1+ 1)(r2+ 1)

where, in this case,∆(Σd) denotes the sum of the differences between the values of defor the two split edges and the values of deof the new edges after the split.

• • (a) • • (b) • • (c)

Figure 11:Edge insertion where (a)JNT remains the same, (b)JNT increases by 1, (c)JNT increases by 2.

3.5.2. Merging two T-meshes

We consider two T-meshes∆1and∆2having a common boundary segment. The new T-mesh∆1∪∆2is

ob-tained by the union of the sets of cells of∆1and∆2. In the following, we assume that (1) and (2) always hold, both

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• • 0 1 2 1 2 Pn1 u1,v1[0, 1]⊗ Pn2 u2,v2[0,1] Pn1 u1,v1[0, 1]⊗ Pn2 u2,v2[1,2] Pn1 u1,v1[1, 2]⊗ Pn2 v2,w2[0,1] Pn1 u1,v1[1, 2]⊗ Pn2 u2,w2[1,2] (a) • • 0 1 2 1 2 Pn1 u1,v1[0, 1]⊗ Pn2 u2,v2[0,1] Pn1 u1,v1[0, 1]⊗ Pn2 u2,v2[1,2] Pn1 u1,v1[1, 2]⊗ Pn2 v2,w2[0,1] Pn1 u1,v1[1, 2]⊗ Pn2 u2,w2[1,2] (b)

Figure 12: Examples where a composite edge for whichde= 0 is split, by the edge insertion (represented by the thick black segment), into two composite edges for whichde= 1(in (a)JNT increases by 2, while in (b)JNT increases by 1).

First, we observe that

N(∆1∪∆2) = N(∆1) + N(∆2).

Let us denote by Wb

1 the number of vertices of∆1along the common boundary which are not corner vertices, and

by Wb

2 the same quantity for∆2(such vertices are not T-junctions, respectively in∆1and∆2, since they are on the

boundary). We denote instead by WIthe number of the common boundary vertices which are not corner vertices. So there are W1b+Wb

2+ 2 −WIvertices which, after the merging, become T-junctions and then

JNT(∆1∪∆2) = JNT(∆1) + JNT(∆2) − (W1b+W2b+ 2 −WI).

There are W1b+ 1 edges of ∆1 on the boundary segment in common with∆2, and W2b+ 1 edges of ∆2 on the

boundary segment in common with∆1. These edges are composite edges for which de= 2, since the vertices on the boundary of a T-mesh are not considered T-junctions. After having merged the two T-meshes, on the common boundary there are WI vertices which are not T-junctions, which means that there are WI+ 1 composite edges,

which can have different values of de: let us say that EIi of them are composite edges with de= i, for i = 0, 1, 2, and so EI0+ E1

I+ EI2= WI+ 1. If the common boundary segment is horizontal, then we have Ehor2 (∆1∪∆2) = Ehor2 (∆1) + Ehor2 (∆2) − (W1b+W2b+ 2 − EI2),

Ehori (∆1∪∆2) = Ehori (∆1) + Ehori (∆2) + EIi, i= 0, 1, and, as a consequence, the new dimension of is :

dim(GSn,r

u,v(∆1∪∆2)) = dim(GSun,v,r(∆1)) + dim(GSnu,r,v(∆2))

− (r1+ 1)(r2+ 1)(W1b+W2b+ 2 −WI)

− (r2+ 1)(n1− 2r1− 1)(W1b+W2b+ 2 − EI2)

+ (r2+ 1)(n1− 2r1− 2)EI1+ (r2+ 1)(n1− 2r1− 3)EI0. If the common boundary segment is vertical, we have

Ever2 (∆1∪∆2) = Ever2 (∆1) + Ever2 (∆2) − (W1b+W2b+ 2 − EI2), Everi (∆1∪∆2) = Everi (∆1) + Everi (∆2) + EIi, i= 0, 1,

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and then

dim(GSnu,v,r(∆1∪∆2)) = dim(GSun,r,v(∆1)) + dim(GSnu,v,r(∆2))

− (r1+ 1)(r2+ 1)(W1b+W2b+ 2 −WI) − (r1+ 1)(n2− 2r2− 1)(W1b+W2b+ 2 − EI2) + (r1+ 1)(n2− 2r2− 2)EI1+ (r1+ 1)(n2− 2r2− 3)EI0 • • • • • • ◦ • ◦ • ◦ • • • • • ◦ • • • • • • ◦ ◦ ◦

Figure 13: Example of two merged T-meshes (common boundary represented by the thick black segment)

4. Approximation power

This section is devoted to the study of the approximation properties of the generalized spline spaces over T-meshes. We will prove these properties for the case where the couples of nonpolynomial functions uRand vRare the same in each cell R, that is, uR= (uR

1, uR2) = (u1, u2) = u and vR= (vR1, vR2) = (v1, v2) = v for any R ∈∆.

More-over, we will assume that(u1, u2) and (v1, v2) give a space Pun,v([minR∈∆aR, maxR∈∆bR]× [minR∈∆cR, maxR∈∆dR]) invariant under translations. More precisely, we assume that, for any(s0,t0) ∈ IR2,

ψ(s,t) ∈ Pn u,v([min

R∈∆aR, maxR∈∆bR] × [minR∈∆cR, maxR∈∆dR]) =⇒ψ(s − s0,t − t0) ∈ Pun,v([min

R∈∆aR, maxR∈∆bR] × [minR∈∆cR, maxR∈∆dR]), (25)

or, equivalently, ψ(s) ∈ Pn1 u1,v1([minRaR, maxRbR]) =⇒ ψ(s − s0) ∈ P n1 u1,v1([minRaR, maxRbR]), ψ(t) ∈ Pn2 u2,v2([minRcR, maxRdR]) =⇒ ψ(t − t0) ∈ P n2 u2,v2([minRcR, maxRdR]).

In order to better understand what this assumption actually means, we observe that the results in [22] (see Section 3) imply that a space of type Pun,v([a, b]), n ≥ 2, invariant under translations must satisfy

ψ(s) ∈ Pn

u,v([a, b]) =⇒ψ′(s) ∈ Pun,v−1([a, b]). (26)

By using elementary arguments of the theory of ordinary differential equations, we obtain that, in order to satisfy (26) (and (2)-(3) as well), both u1, v1and u2, v2must be chosen in one of the following ways:

• u(s) = eλs, v(s) = eµs, withλ,µ∈ IR,λ6=µ;

• u(s) = eλs, v(s) = seλs;

• u(s) = eαscoss), v(s) = eαssins), withα,β∈ IR andβ(b − a) <π.

It can be easily verified that with any of the above choices the corresponding space is invariant under transla-tions. As a consequence, the assumption (25) is equivalent to choosing(u1, u2) and (v1, v2) as mentioned above.

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Remark 3. It is easy to verify that all the possible choices of u and v reported above satisfy not only (26) but also ψ(s) ∈ Pn u,v([a, b]) =⇒ Z ψ(s)ds ∈ Pn+1 u,v ([a, b]). (27)

This leads to generalized spline spaces GSun,r,v(∆) satisfying

ψ(s,t) ∈ GSn,r u,v(∆) =⇒ Dsψ(s,t) ∈ GSu˜n,vs,˜rs(∆) ψ(s,t) ∈ GSn,r u,v(∆) =⇒ Z ψ(s,t)ds ∈ GSˆns,ˆrs u,v (∆) ψ(s,t) ∈ GSn,r u,v(∆) =⇒ Dtψ(s,t) ∈ GSu˜nt,v,˜rt(∆) ψ(s,t) ∈ GSn,r u,v(∆) =⇒ Z ψ(s,t)dt ∈ GSˆnt,ˆrt u,v (∆), where ˜ns= (n1− 1, n2), ˆns= (n1+ 1, n2), ˜nt= (n1, n2− 1), ˆnt= (n1, n2+ 1), and ˜rs= (r1− 1, r2), ˆrs= (r1+

1, r2), ˜rt= (r1, r2− 1), ˆrt= (r1, r2+ 1). In other words, we get spaces whose elements have derivatives and

integrals belonging to spaces of the same type. Such nice behaviour with respect to the fundamental derivation and integration operators is of a certain interest in some applications, in particular in isogeometric analysis (see, e.g., [13], [14], [18]). Moreover, we observe that noteworthy cases of generalized spline spaces allowing to exactly reproduce certain shapes (conic sections, helices, cycloids, catenaries; see also [18]), such as u(s) = cos(βs),

v(s) = sin(βs) and u(s) = cosh(λs), v(s) = sinh(λs), satisfy the invariance under translations.

We will obtain the approximation order by using similar arguments to the ones used in [1], and introducing a new suitable quasi-interpolant operator. In fact, the local approximants used in [1], that is, the averaged Taylor expansions, cannot be simply generalized to our non-polynomial case. Moreover, also the results on the approxi-mation power obtained in [6] for the univariate case, by using Hermite interpolation in spaces of type Pun,v([a, b]),

cannot be directly extended to the bivariate case, due to the difficulty to find a suitable differential operator and the corresponding Green’s function needed to construct a non-polynomial Taylor expansion. For these reasons, we adopt an alternative approach: we construct a bivariate Hermite interpolant belonging to the spline space, whose existence is rigorously proved by using the assumption (2) and (3). This also allows us to obtain an approximation order, which is essentially the same as in polynomial case.

Given a function f ∈ Cn+1() and (s

0,t0) ∈ (a, b) × (c, d), we define the interpolant QL( f ; s0,t0)(s,t) as the

func-tion satisfying the two following condifunc-tions 1. it belongs to Pun,v([a, b] × [c, d]),

2. its polynomial expansion of coordinate bi-degree(n1, n2) coincides with the polynomial expansion of f of

the same bi-degree, that is, QL( f ; s0,t0)(s,t) is a Hermite interpolant of coordinate bi-degree (n1, n2).

Since QL( f ; s0,t0) is a Hermite interpolant, the Taylor expansion of the difference f − QL( f ; s0,t0) does not contain

any term of degree smaller than or equal to k, where k := min{n1, n2}, and then k f − QL( f ; s0,t0)k = O(hk+1),

where h := diam([a, b] × [c, d]).

In order to show that QL( f ; s0,t0)(s,t) exists and is unique for any f ∈ Cn+1(Ω) and (s0,t0) ∈ (a, b) × (c, d),

let us write the explicit expressions of a generic element belonging to Pun,v([a, b] × [c, d])

n1−2

i=0 n2−2

j=0 ai j (s − s0)i i! (t − t0)j j! + n1−2

i=0 bi (s − s0)i i! u2(t) + n1−2

i=0 ci (s − s0)i i! v2(t) + n2−2

j=0 dju1(s) (t − t0)j j! + n2−2

j=0 ejv1(s) (t − t0)j j! +ν1u1(s)u2(t) +ν2u1(s)v2(t) +ν3v1(s)u2(t) +ν4v1(s)v2(t)

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and of its Taylor expansion of coordinate bi-degree(n1, n2) ∑n1−2 i=0 ∑ n2−2 j=0 ai j(s−s0) i(t−t 0)j i! j! +∑ n1−2 i=0 ∑ n2 j=0 biDtju2(t0) i! j! (s − s0)i(t − t0)j +∑n1−2 i=0 ∑ n2 j=0 ciDtjv2(t0) i! j! (s − s0) i(t − t 0)j+∑ni=01 ∑ n2−2 j=0 djDisu1(s0) i! j! (s − s0) i(t − t 0)j +∑n1 i=0∑ n2−2 j=0 ejDisv1(s0) i! j! (s − s0)i(t − t0)j +ν1∑ni=01 ∑ n2 j=0 Di su1(s0)Dtju2(t0) i! j! (s − s0) i(t − t 0)j +ν2∑ni=01 ∑ n2 j=0 Disu1(s0)Dtjv2(t0) i! j! (s − s0)i(t − t0)j +ν3∑ni=01 ∑ n2 j=0 Di sv1(s0)Dtju2(t0) i! j! (s − s0) i(t − t 0)j +ν4∑ni=01 ∑ n2 j=0 Disv1(s0)Dtjv2(t0) i! j! (s − s0)i(t − t0)j.

Then, the condition requiring that QL( f ; s0,t0) is a Hermite interpolant of coordinate bi-degree (n1, n2) corresponds

to the following equations:

ai j+ biDtju2(t0) + ciDtjv2(t0) + djDisu1(s0) + ejDisv1(s0) +ν1Disu1(s0)Dtju2(t0) +ν2Disu1(s0)Dtjv2(t0) +ν3Disv1(s0)Dtju2(t0) +ν4Disv1(s0)Dtjv2(t0) = DisD j tf(s0,t0), for 0≤ i ≤ n1− 2, 0 ≤ j ≤ n2− 2, biDtju2(t0) + ciDtjv2(t0) +ν1Disu1(s0)Dtju2(t0) +ν2Disu1(s0)Dtjv2(t0) +ν3Disv1(s0)Dtju2(t0) +ν4Disv1(s0)Dtjv2(t0) = DisD j tf(s0,t0), for 0≤ i ≤ n1− 2, and j = n2− 1, n2, djDisu1(s0) + ejDisv1(s0) +ν1Disu1(s0)Dtju2(t0) +ν2Disu1(s0)Dtjv2(t0) +ν3Disv1(s0)Dtju2(t0) +ν4Disv1(s0)Dtjv2(t0) = DisD j tf(s0,t0), for i= n1− 1, n1, 0≤ j ≤ n2− 2, and ν1Disu1(s0)Dtju2(t0) +ν2Disu1(s0)Dtjv2(t0) +ν3Disv1(s0)Dtju2(t0) +ν4Disv1(s0)Dtjv2(t0) = DisD j t f(s0,t0),

for i= n1− 1, n1, j= n2− 1, n2. By using a suitable reordering of the unknowns ai j, bi, ci, dj, ejk, we obtain a linear system whose matrix is

A=     I ⋆ ⋆ ⋆ 0 A1 0 ⋆ 0 0 A2 ⋆ 0 0 0 A3    

where I is the identity matrix of size(n1− 1)(n2− 1) × (n1− 1)(n2− 1), ⋆ stands for blocks of suitable size, 0

stand for null matrices of suitable size, and

A1=             Dn2−1 t u2(t0) 0 . . . 0 Dtn2−1v2(t0) 0 . . . 0 Dn2 t u2(t0) 0 . . . 0 Dtn2v2(t0) 0 . . . 0 0 Dn2−1 t u2(t0) . . . 0 0 Dtn2−1v2(t0) . . . 0 0 Dn2 t u2(t0) . . . 0 0 Dtn2v2(t0) . . . 0 0 0 . .. 0 0 0 . .. 0 0 0 . . . Dn2−1 t u2(t0) 0 0 . . . Dnt2−1v2(t0) 0 0 . . . Dn2 t u2(t0) 0 0 . . . Dnt2v2(t0)            

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